Almost discrete SV-space. (English) Zbl 0791.54049

Henriksen, M.; Wilson, R. G.

Topology Appl. 46, No. 2, 89-97 (1992).

Summary: A Hausdorff space is called almost discrete if it has precisely one nonisolated point. A Tykhonov space \(Y\) is called an SV-space if \(C(Y)/P\) is a valuation ring for every prime ideal \(P\) of \(C(Y)\). It is shown that the almost discrete space \(X= D\cup\{\infty\}\) is an SV-space if and only if \(X\) is a union of finitely many closed basically disconnected subspaces if and only if \(M_ \infty=\{f\in C(X): f(\infty)= 0\}\) contains only finitely many minimal prime ideals. Some unsolved problems are posed.

MSC:

54G05	Extremally disconnected spaces, \(F\)-spaces, etc.
54C40	Algebraic properties of function spaces in general topology

Keywords:

almost discrete space; SV-space

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